Wind-current feedback is an energy sink for oceanic internal waves

Internal waves contain a large amount of energy in the ocean and are an important source of turbulent mixing. Ocean mixing is relevant for climate because it drives vertical transport of water, heat, carbon and other tracers. Understanding the life cycle of internal waves, from generation to dissipation, is therefore important for improving the representation of ocean mixing in climate models. Here, we provide evidence from a regional realistic numerical simulation in the northeastern Pacific that the wind can play an important role in damping internal waves through current feedback. This results in a reduction of 67% of wind power input at near-inertial frequencies in the region of study. Wind-current feedback also provides a net energy sink for internal tides, removing energy at a rate of 0.2 mW/m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^2$$\end{document}2 on average, corresponding to 8% of the local internal tide generation at the Mendocino ridge. The temporal variability and modal distribution of this energy sink are also investigated.


Wind and surface Currents spectra
To complement the co-spectrum of wind and currents presented in Fig. 1 of the main document, we provide here the individual spectra of winds and surface currents (Fig. S2) Figure S2: Horizontal wavenumber-frequency spectra of (a) winds and (b) surface currents. This figure has been generated using Python v3.6https://www.Python.org/

Wind power bulk formula
We write u w the wind vector and u c the ocean current vector, with u w = |u w | and u c = |u c |. If we define a normal base (x c , y c ) where x c is aligned with u c and y c is the unit vector orthogonal to x c , then we can rewrite the two vectors in this base as: where θ is the relative angle between the wind and the current.
The wind power can be rewritten as a function of the wind and current magnitude and of the relative angle between the two, F (u w , u c , θ) = τ · u c . In the case of a constant wind, and of an ideally polarized current, with currents oscillating along a single direction, the average wind power over one period is shown in Fig. S3. The energy sink associated with wind power is maximal when and where the wind is aligned with the currents and minimal when the wind is at right angle of the current. Figure S3: Wind power as a function of θ, the relative angle between the wind and the current (polar axis) for values of u w =10 ms −1 , u c =1 ms −1 , and a constant C d = 0.0012. This figure has been generated using Python v3.6 https://www.Python.org/

Internal tide energy flux
The internal tide energy flux is computed as: where H is the water depth, ζ is the sea surface height, (u ′ , v ′ ) is the baroclinic horizontal velocity associated with internal tide and p ′ is the pressure perturbation associated with internal tides. The overbars correspond to a one-year average and the primes correspond to a combination of a temporal Butterworth band-pass filter with cutoff periods at 4 and 14 hours, and a spatially uniform highpass filter with a cutoff scale of 180 km. Figure S4: Intensity (color) and direction (arrows) of the depth-integrated baroclinic energy flux for internal tides. Bathymetry is in countours. This figure has been generated using Python v3.6https://www.Python.org/ The vertically-integrated baroclinic internal tide energy flux shows a beam of internal emanating from the Mendocino Ridge at 40 • N and propagating southward, as well as incoming beams propagating eastward from the northern boundary.

Baroclinic modes decomposition
The Stourm-Liouville equation for normal modes with a free surface boundary condition (e.g., Kelly, 2016) is solved at each grid point using a seasonal time averaged stratification to obtain a set of eigenfunctions. An example of these eigenfunctions is given in Fig. S5.
The decomposition of the velocity on baroclinic modes shows that on average more than 70% of the surface kinetic energy is explained by the superposition of the first 3 baroclinic modes (Fig. S6).